I always took it as her reminding Tiana that a part of her father lives on in her. She specifically mentions about the father caring for his family too, so maybe this is also her way of telling Tiana that she will need to look out for her family too (such as her mother and future husband).

Something like this. I think the takeaway is that the restaurant isn't/shouldn't be the most important thing in her life -- it's the people she loves.

^